Analytical Study of Electronic Structure in Armchair Graphene Nanoribbons

We present the analytical solution of the wave function and energy dispersion of armchair graphene nanoribbons (GNRs) based on the tight-binding approximation. By imposing the hard-wall boundary condition, we find that the wave vector in the confined direction is discretized. This discrete wave vector serves as the index of different subbands. Our analytical solutions of wave function and associated energy dispersion reproduce the results of numerical tight-binding and the solutions based on the $\mathbf{k}∙\mathbf{p}$ approximation. In addition, we also find that all armchair GNRs with edge deformation have energy gaps, which agrees with recently reported first-principles calculations.